Coherent and convex risk measures for portfolios with applications

Wei, Linxiao; Hu, Yijun

doi:10.1016/j.spl.2014.03.005

Cited by 25 publications

(15 citation statements)

References 13 publications

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“…e delicate issue however is what family of multivariate risk measures should be used. It turns out that to produce a quasi-concave or coherent acceptability index for portfolio vectors, one needs to make an adaptation to the definition of coherent and convex risk measures for portfolio vectors introduced by Burgert and Rüschendorf [6]; see also Rüschendorf [10] and Wei and Hu [11]. Definition 2 is such an adaptation.…”

Section: Definitionmentioning

confidence: 99%

“…It should be mentioned that there also exist many papers about multidimensional risk measures. For scalar multivariate risk measures, see Burgert and Rüschendorf [6], Rüschendorf [7], Ekeland and Schachermayer [8], Ekeland et al [9], Rüschendorf [10], Wei and Hu [11], Chen et al [12], and the references therein. For set-valued multivariate risk measures, see Jouini et al [13], Hamel [14], Hamel and Heyde [15], Hamel et al [16], Hamel et al [17], Labuschagne and Offwood-Le Roux [18], Farkas et al [19], Molchanov and Cascos [20], Chen and Hu [21], and the references therein.…”

Section: Introductionmentioning

confidence: 99%

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Acceptability Indexes for Portfolio Vectors

Zeng

Chen

2019

Mathematical Problems in Engineering

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In this paper, we introduce two new classes of acceptability indexes, named quasi-concave acceptability indexes and coherent acceptability indexes, for portfolio vectors. We establish the one-to-one correspondence between quasi-concave (coherent, resp.) acceptability indexes and convex (coherent, resp.) risk measures for portfolio vectors. We derive the representation results for coherent and convex risk measures. Finally, based on these results, we derive the representation results for quasi-concave acceptability indexes and coherent acceptability indexes for portfolio vectors. These new acceptability indexes can be considered as a kind of multivariate extension of univariate coherent and quasi-concave acceptability indexes introduced by Cherny and Madan (2009) and Rosazza Gianin and Sgarra (2013), respectively.

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Section: Definitionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Acceptability Indexes for Portfolio Vectors

Zeng

Chen

2019

Mathematical Problems in Engineering

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“…Com base no conceito de medida de risco de máxima correlação, representação dual de medidas convexas com lei invariante são determinadas, do mesmo jeito que são baseadas no AVaR para o caso univariado. Wei & Hu (2014) complementam essa abordagem considerando casos independentes de modelo, convertendo axiomas, conjunto de aceitação e representação dual.…”

Section: Extensões: Abordagem Multivariadaunclassified

Teoria de Medidas de Risco: uma revisão abrangente

Righi

Ceretta

2014

Braz. Rev. Fin.

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A fundamental aspect of proper risk management is the measurement, especially forecasting of risk measures. Measures such as variance, volatility and Value at Risk had been considered valid because of their practical intuition. However, a solid theoretical framework it is important to ensure better properties for risk measures. Such background is the risk measures theory. This paper presents a comprehensive literature review on risk measures theory, focusing in basic theory and extensions to this fundamental outline. The paper is structured in order to cover the main risk measures classes from literature, which are coherent risk measures, convex risk measures, spectral and distortion risk measures and generalized deviation measures.

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“…In [12], the authors first introduced the scalar multivariate coherent and convex risk measures, see also [13]. For more works on multivariate risk measures, see [8,9,[14][15][16][17] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Multivariate Shortfall and Divergence Risk Statistics

Song

Zeng

Chen

et al. 2019

Entropy

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The aim of this paper is to construct two new classes of multivariate risk statistics, and to study their properties. We, first, introduce the multivariate shortfall risk statistics and multivariate divergence risk statistics. Then, their basic properties are studied, and their representation results are provided. Furthermore, their coherency is also characterized by means of the corresponding loss function. Finally, entropic risk statistics are given to illustrate the proposed new classes of multivariate risk statistics. The relationship between multivariate shortfall and divergence risk statistics is also discussed.The purpose of the present paper is to construct multivariate shortfall and divergence risk statistics, respectively, and to study their properties, including their representation results. We provide the representation results for the multivariate shortfall and divergence risk statistics with explicit penalty functions, which are expressed in terms of the corresponding loss functions or divergence functions, respectively. The coherency of the univariate shortfall risk statistics is also characterized by means of the corresponding loss function. Finally, as examples, the multivariate entropic (or entropy-like) risk statistics are constructed to illustrate the proposed multivariate shortfall and divergence risk statistics.The steps and methods of the present paper are as follows. We, first, introduce the acceptance set of the accepted portfolios. Then, the multivariate shortfall risk statistics are introduced, and their properties including their representations are investigated. Further, the coherency of the univariate shortfall risk statistics is characterized. Meanwhile, a kind of multivariate risk statistic closely related the multivariate shortfall risk statistic is also introduced and investigated, which is the so-called multivariate divergence risk statistic. Finally, examples are given. Convex analysis is employed to complete the involved argumentation.The main contribution of the present paper are as follows. First, we have constructed two new meaningful classes of multivariate risk statistics, which are multivariate shortfall and divergence risk statistics. Second, their properties including the representation results are investigated, and their coherency is characterized. Finally, multivariate entropic (or entropy-like) risk statistics are introduced.The rest of the paper is organized as follows. In Section 2, we briefly state some preliminaries including the definitions of multivariate shortfall and divergence risk measures. The main results are stated in Section 3. In Section 4, All the proofs of the main results of the paper are provided. In Section 5, examples are given. Finally, conclusions are summarized.

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Coherent and convex risk measures for portfolios with applications

Cited by 25 publications

References 13 publications

Acceptability Indexes for Portfolio Vectors

Acceptability Indexes for Portfolio Vectors

Teoria de Medidas de Risco: uma revisão abrangente

Multivariate Shortfall and Divergence Risk Statistics

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